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14 IEEE Control Systems Magazine October 20010272-1708/01/$10.00©2001IEEE

SPECIAL SECTIONSPECIAL SECTION

By Peter Young and Arun Chotai

In the environmental and agricultural sciences, as in many ar-eas of engineering and science, mathematical models are nor-mally formulated as deterministic differential (or equivalentdiscrete-time) equations. Most often, the structure of theseequations is defined by the scientist or engineer in a form thatis perceived to be appropriate according to the physical na-

ture of the system and the current scientific paradigms in the area ofstudy. Most of the “simulation” models that emerge from this ap-proach are very large, with many unknown parameters. Conse-quently, they are difficult, if not impossible, to identify, estimate, andvalidate in rigorous statistical terms because of problems associatedwith over-parametrization and the lack of experimental or monitoreddata. Unlike the situation in engineering, however, natural environ-mental systems, and many systems in agriculture, are not normally

Young ([email protected]) and Chotai are with the Centre for Researchon Environmental Systems and Statistics, Institute of Environmental and Natu-ral Sciences, Lancaster University, Lancaster LA1 4YQ, U.K. ©

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Authorized licensed use limited to: Lancaster University Library. Downloaded on January 22, 2009 at 06:48 from IEEE Xplore. Restrictions apply.

manmade. As a result, their internal physical, biological,and ecological mechanisms are often poorly understood,and planned experiments that might lead to improvementsin such understanding are either difficult or, in the case ofenvironmental systems, even impossible to undertake.

Increasingly in recent years, however, the poorly definednature of many environmental and agricultural systems hasbeen recognized. It is now becoming accepted that a newmodeling philosophy and associated methodology is re-quired that acknowledges the need to quantify the uncer-tainty associated with the model (e.g., [1]-[3]). The Centre forResearch on Environmental Systems and Statistics (CRES) atLancaster University has been prominent in this area of re-search. This article outlines the major aspects of our ap-proach to stochastic modeling, as well as briefly describingthree studies that demonstrate the practical utility of this ap-proach. In particular, the article shows how parametricallyefficient (parsimonious), low-order stochastic models can beproduced that reflect the dominant modal characteristics ofthe system and can be interpreted in physically meaningfulterms. Within the present context, the most important aspectof these data-based mechanistic (DBM) models is that theyprovide an appropriate basis for advanced nonstationary ornonlinear signal processing, adaptive forecasting, andmultivariable control system design.

Unlike data-based “black-box” models, DBM models arerequired to have a mechanistic interpretation. Indeed, theyare not deemed truly credible in scientific terms unless suchan interpretation proves possible. But, unlike the situationwith “grey-box” models, such a physically meaningful expla-nation is not imposed as a hypothesis prior to modeling.Rather, it emerges from an inductive modeling procedureonly after the model structure has first been identified parsi-moniously from a generic class of models with wide applica-tion potential. In this manner, prior scientific prejudiceabout the nature and complexity of the model is avoided,and the resulting DBM model will normally have a structureand parametrization that is appropriate to the informationcontent of the data. In this article, the generic class of mod-els used in DBM modeling comprises stochastic linear andnonlinear differential or difference equations, or their trans-fer function equivalents.

DBM modeling methods were developed originally forthe analysis of measured time-series data, particularly in re-lation to the design of signal processing, forecasting, and au-tomatic control systems. In the early stages of modeling,when observational data are either scarce or not available,however, the same methodological tools can also be used toobtain reduced-order versions of the large simulation mod-els. The importance of reduced-order models that reflectthe dominant modes of system behavior is being widely rec-ognized in many areas of application within the environmen-tal and agricultural sciences. For example, such modelshave been used in the simplification of large simulationmodels used in climate research (e.g., [4]) and in optimal

control system design (e.g., [5]). An example of the DBM ap-proach to model reduction is the modeling and control ofthe microclimate in a large horticultural greenhouse [6]where the reduced-order control model is obtained from amuch larger nonlinear simulation model. A similar DBM ap-proach has been used to obtain linear, reduced-order ver-sions of the topically important “global carbon cycle”models [6], [7] used in the studies of the International Panelon Climate Change (IPCC).

This article outlines the main aspects of DBM modeling,as well as the associated methods of signal processing,adaptive forecasting, and multivariable control that exploitsuch models. This overall generic approach appears to havewide application potential, and its practical utility is illus-

October 2001 IEEE Control Systems Magazine 15

Table of Abbreviations

AMV Active mixing volume

AR Autoregressive (model)

ARC Adaptive radar calibration (system)

RT2 Coefficient of determination based on

simulation response error

CRES Centre for Research on EnvironmentalSystems and Statistics

CV Control volume

DBM Data-based mechanistic

DMA Dominant mode analysis

FIS Fixed-interval smoothing

GPC Generalized predictive control

GSA Generalized sensitivity analysis

HMC Hybrid-metric-conceptual (model)

IPCC International Panel on Climate Change

KF Kalman filter

LQ Linear quadratic

LQG Linear quadratic Gaussian

MFD Matrix fraction description

MCS Monte Carlo simulation

NMSS Nonminimal state space

PDF Probability distribution function

PIP Proportional-integral-plus (controller)

RLS Recursive least-squares (method)

RIV Refined instrumental variable (method)

SRI Silsoe Research Institute

SDP State-dependent parameter (model)

SVF State variable feedback (control)

SRIV Simplified refined instrumental variable(method)

TF Transfer function (model)

TVP Time-variable parameter (model)

UC Unobserved component (model)


trated by three practical examples. These range from themodel reduction and control application mentioned abovethrough the modeling and control of forced ventilation sys-tems in agricultural buildings to the design of adaptive floodforecasting and warning systems.

Methodological BackgroundThe methods used in this article have been developed overthe past 20 years, and they cover four main areas: DBMmethods for modeling linear and nonlinear stochastic sys-tems; dominant mode analysis (DMA) leading to the com-bined linearization and simplification of large simulationmodels; unobserved component (UC) models and their usein the recursive estimation, forecasting, and smoothing ofnonstationary time series; and nonminimal state space(NMSS) methods of control system design. All of these meth-ods have been described in detail elsewhere, so they areoutlined only briefly in the following subsections.

Data-Based Mechanistic ModelingPrevious publications (e.g., [6]-[10]) illustrate the evolutionof the DBM philosophy and its methodological underpin-ning. This methodological basis is heavily dependent on theexploitation of recursive estimation in all its forms. Theseinclude the traditional application to online and offline pa-rameter estimation using recursive least-squares (RLS) andoptimal refined instrumental variable (RIV) algorithms; thedesign of adaptive forecasting systems using the Kalman fil-ter (KF) algorithm; the use of recursive fixed-intervalsmoothing (FIS) for more statistically efficient, offlinetime-variable parameter (TVP) estimation for use in signalextraction; and the exploitation of FIS combined with otherprocedures for state-dependent parameter (SDP) estima-tion and the nonparametric/parametric identification ofnonlinear stochastic systems.

The three major phases in the DBM modeling strategyare as follows:

• Stochastic Simulation Modeling and Sensitivity Analysis(e.g., [2], [3], [6], [8], and the references therein). Inthe initial phases of modeling, observational data maywell be scarce, so any major modeling effort will haveto be centered on simulation modeling, usually basedinitially on largely deterministic concepts such as dy-namic mass and energy conservation. Recognizingthe inherent uncertainty in many environmental andagricultural systems, however, these deterministicsimulation equations can be converted into an alter-native stochastic form. Here, it is assumed that the as-sociated parameters and inputs can be represented insome suitable stochastic form such as a probabilitydistribution function (PDF) for the parameters andtime-series models for the inputs. The subsequentstochastic analysis then exploits Monte Carlo simula-tion (MCS) in three ways: first, to explore the propaga-tion of uncertainty in the resulting stochastic model;

second, as a mechanism for generalized sensitivityanalysis (GSA) to identify the most important parame-ters leading to a specified model behavior; and third,the use of MCS in stochastic optimization.

• Dominant Mode Analysis and Simulation Model Simpli-fication (e.g., [6], [7], and the references therein). Theinitial exploration of the simulation model in stochas-tic terms can reveal the relative importance of differ-ent parts of the model in explaining the dominantbehavioral mechanisms. This understanding of themodel is further enhanced by employing a novelmethod of combined statistical linearization andmodel order reduction. This is applied to time-seriesdata obtained from planned experimentation, not onthe system itself, but on the simulation model that be-comes a surrogate for the real system. Such DMA is ex-ploited to develop low-order, dominant modeapproximations of the simulation model; approxima-tions that are often able to explain its dynamic re-sponse characteristics to a remarkably accuratedegree (e.g., coefficients of determination > 0.999; i.e.,greater than 99.9% of the high-order model output isexplained by the low-order model). Conveniently, thestatistical methods used for such linearization and or-der reduction exercises are the same as those used forthe next phase in the modeling process.

• DBM Modeling from Real Data (see previous refer-ences). Time-series data form the basis for stochasticDBM modeling. The DBM models characterize thosedominant modes of the system behavior that areclearly identifiable from the time-series data, and un-like simulation models, the efficacy of the DBM mod-els is heavily dependent on the quality of these data.The models derived in this form normally constitutethe main vehicles for the subsequent design of fore-casting and automatic control systems. In the case ofconstant-parameter models, the methodologicaltools used in this final stage of DBM analysis are linear,constant parameter, transfer function (TF) identifica-tion and estimation tools, such as optimal instrumen-tal variable methods (SRIV/RIV; e.g., [11] and thereferences therein). These can be compared with thealternative and better known methods, such as thosein the MATLAB Identification Toolbox, which are de-signed specifically for backward shift operator mod-els and, in most cases, require concurrent estimationof noise model parameters (note that the RIV/SRIV al-gorithms are different from the IV/IV4 options in theIdentification Toolbox). In contrast, RIV/SRIV algo-rithms are available for the estimation of TF modelsin all the major operators (backward shift, time de-rivative, and delta) and do not require concurrent es-timation of a noise model. If the system underinvestigation is nonstationary, however, then the con-stant-parameter estimation is replaced by TVP esti-

16 IEEE Control Systems Magazine October 2001


mation [12]-[14], whereas if the system is heavilynonlinear, the SDP estimation allows for identificationof the nonlinear model structure prior to final nonlin-ear parametric estimation (see [7], [9], [14]). This lat-ter approach provides an alternative to, or areinforcement of, other approaches to nonlinear mod-eling, such as neural network and neuro-fuzzy model-ing (e.g., [15]). It has the advantage that the resultingmodels are normally much less complex, of consider-ably lower order, and more easily interpretable inmechanistic terms.

DBM modeling is obviously very different from physi-cally based simulation modeling, but it is often confusedwith grey-box modeling. Grey-box modeling follows the tra-ditional hypothetico-deductive approach. Here the hypoth-esis is normally in the form of a much-simplified model ofthe physical system under study, and deduction is based onestimating the parameters that characterize this assumedmodel structure from measured data. In contrast, DBM mod-eling is inductive and commences with few prior percep-tions of the model form, except that it is a member of asuitable, generic class of models (normally linear/nonlineardifferential equations or their discrete-time equivalents).Methods of statistical model structure identification and pa-rameter estimation are then utilized to produce a low-di-mensional, black-box model, within this generic class ofmodels, that explains the data unambiguously in a statisti-cally efficient manner. Only then is the question of the un-derlying, mechanistic interpretation of the modeladdressed, as illustrated by the examples discussed later. Inthis manner, a minimally parametrized and statisticallywell-defined model is obtained, and the dangers of imposingtoo much confidence in prior assumptions about the physi-cal nature of the system are avoided.

Signal Processing, Forecasting,and Automatic ControlThe DBM methods were developed primarily for modelingsystems from normal observational time-series data ob-tained from monitoring exercises (or planned experimen-tation, if possible) carried out on the real system. Often,such time series require some form of preprocessing, andthis is accomplished using methods of nonstationarytime-series analysis based on recursive fixed-intervalsmoothing (e.g., [11], [12], and the references therein).These powerful statistical tools allow for interpolationover gaps in the time series; the detection of outliers; “sig-nal extraction,” including the estimation and removal ofperiodic components (e.g., seasonal adjustment); andtime-frequency analysis.

The DBM models are also in a form appropriate formodel-based forecasting and control system design. Ofcourse, any available methods can be used for these appli-cations. At Lancaster, however, the stochastic UC model-ing approach based on TVP estimation provides the main

vehicle for adaptive forecasting ([12], [13], and thereferences therein). Model-based automatic control sys-tem design is formulated within the NMSS setting, normallyresulting in the multivariable proportional-integral-plus(PIP) control algorithm (see [16] and the referencestherein). The NMSS is the most natural state-space defini-tion for discrete-time transfer function models, and the PIPcontroller provides a natural multivariable extension ofthe conventional PI and PID controllers. In addition to pro-portional and integral action, it includes additional for-ward path and feedback filters that allow for NMSS statefeedback, without the need for state reconstruction, thusenhancing robustness and closed-loop performance. Ofcourse, the PIP controller is just one specific outcome ofthe NMSS design concept. This concept is not only attrac-tive in its own right, but control algorithms that derivefrom it are very general in form and thus able to mimicother well-known design procedures, such as minimalstate linear quadratic Gaussian (LQG), generalized predic-tive control (GPC), and Smith predictor control for time-de-lay systems (see [17] and the references therein).

It should be noted that many of the time-series analysisand modeling procedures mentioned above and used inthe examples described below are contained in theMATLAB CAPTAIN time-series analysis and forecastingtoolbox, currently in the final stages of beta testing (seehttp://www.es.lancs.ac.uk/cres/captain/).

Modeling and Control of aGreenhouse MicroclimateThe literature on modeling of the microclimate in green-houses is very large, with models ranging from high-order,nonlinear simulation models (e.g., [18]) through much sim-pler but still physically meaningful data-based models (e.g.,[19]) to purely black-box models (e.g., [5]). One importantaspect of this research has been the development and use ofmodels for the control of greenhouse microclimates (e.g.,[20] and the references therein). This example falls into thelatter category and is concerned with the modeling and au-tomatic control of the microclimate in the large Venlo horti-cultural greenhouse at the Silsoe Research Institute (SRI) inBedford, U.K.

Prior to planned experimentation on the actual green-house at SRI, the control model was obtained by combinedlinearization and order reduction analysis [6], [7], [21]. Thiswas applied to the high-order, nonlinear simulation modelshown in Fig. 1, originally derived by scientists at SRI [18] andconverted to Simulink form at Lancaster. When planned ex-perimentation became possible, these earlier control modelswere replaced by the equivalent TF models obtained by iden-tification and estimation applied to the experimental in-put-output data. In the final stages of the study, the controlmodels were used to design automatic climate control sys-tems that were first evaluated on the full, nonlinear simula-tion model, prior to implementation in the greenhouse.



To simplify the model shown in Fig. 1, it was perturbedabout a number of different operating points by suitably de-signed input functions. These were applied to each controlinput in turn (i.e., the fractional valve aperture of the heat-ing boileru t1, , the input to the mist spraying systemu t2, , andthe CO2 enrichment input u t3, ), and they produced climateperturbations defined by the internal air temperature y t1, in°C, the percentage relative humidity y t2, , and the CO2 con-centration y t3, in ppm. The input-output perturbational dataset obtained in this manner was then used as the basis forthe identification and estimation of reduced-order, dis-crete-time linear models that were directly suitable formodel-based PIP system design [21].

At the most important operating condition, the esti-mated reduced-order model for y t t t t

Ty y y= [ ], , ,1 2 3 , in re-sponse to u t t t t

Tu u u= [ ], , ,1 2 3 , takes the form of thefollowing discrete-time, transfer function matrix model:

y t

zz

z

z=

−−

−−

−

−−

−

0 01431 0 9056

0 0712 0

0 05871 0 79

1

11

1

..

.

.. 06

0 7909 0

001791

1 0 6996831294

1 0

11

1

1

1z

z

zz

z

−−

−

−

−−− −

.

..

..6541 1z

t

−

u .

(1)


CO2

Humidity

Air Temperature

1/s

1/s

1/s

1/s

1/s

1/s

1/s

1/s

1/s

1/s

1/s

×1

×2

×3

×4

×5

×6

×7

×8

×9

×10

×11

WeatherInputs

ControlInputs

Thermal Rad Coeffs

Heat Trans Coeffs

Stomatal Res

CO Model2

Latent Heat En Ex

En Ex

Derivatives

Demux

Mux

u1

u2

u3

u4

u5

u6

u7

u8

Figure 1. Simulink simulation model of the microclimate in the SRI Venlo greenhouse.


This model is well identified, and the parameter estimatesare statistically well defined. As might be expected, the ma-jor coupling is between the temperature and relative humid-ity, with CO2 being almost independent of these variables.Despite its simplicity, the model explains the high-ordernonlinear model response very well, with coefficients of de-termination RT

2 0 99> . (i.e., over 99% of the high-order modelresponse explained by the low-order model). Clearly, thedominant modal behavior of the simulation model at thisoperating condition, which is so important for subsequentcontrol system design, has been captured very well. More-over, this model is much simpler than might be expected onthe basis of the full nonlinear model equations.

From the DBM standpoint, it is important that the re-duced-order model should make good physical sense, and,as expected from mass and energy transfer considerations,the temperature and humidity dynamics are highly coupled.Indeed, it is possible to associate the model (1) with the dif-ferential equations of mass and energy transfer for this sys-tem (much simplified versions of the equations used toderive the high-order model in Fig. 1). Since the model is es-timated in discrete time, however, this relationship is nottoo transparent. Consequently, it is better to delay our illus-tration of such a mechanistic interpretation to the next sec-tion, where a related model is estimated directly incontinuous-time terms and the physical interpretation ismuch more obvious.

The high-order simulation model provided a very use-ful vehicle for initial PIP control system design and evalu-ation exercises [21], where it functioned as a valuablesurrogate for the real system before experimentationwas possible. Furthermore, the fact that the reduced-or-der models were structurally identical to those identifiedlater from the real data meant that the initial PIP control-ler designs were also structurally similar. Note that theinherent stochastic nature of the DBM models is most im-portant in the PIP control system design process, since itallows for evaluation of the controller’s robustness touncertainty using stochastic simulations. These can beeither single simulations with stochastic inputs or MCSused to assess the robustness of the control system touncertainty in the model parameters (e.g., [17] and thereferences therein).

NMSS control system design is straightforward and flexi-ble in this example [21]. Once the reduced-order TF matrixmodel (1) has been obtained, it is converted straightfor-wardly to the left matrix fraction description (MFD), andthence to the following NMSS form:

x Fx Gu Dy y Hxt t t d t t t+ += + + =1 1, (2)

where F and u t are defined as at the bottom of the next pageand

G =

−−

−

0 0143 0 0712 0

0 0587 0 7909 0

0 01791 831294

0 0 0

0 0

. .

. .

. .

0

0 0 0

1 0 0

0 1 0

0 0 1

0 0143 0 0712 0

0 0587 0 7909 0

0 01791 8

−−

−

. .

. .

. 31294.

H =

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

D =

0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 1

T

.

Here x t = − − − − −[ , , , , , , , , ,y y y y y y u u ut t t t t t t t t1 2 3 1 1 2 1 3 1 1 1 2 1 3 − 1

z z zt t tT

1 2 3, , , ] is the NMSS vector; u u u ut t t tT= [ ], , ,1 2 3 is the

control input vector; and y d t d t d t d tTy y y, , , ,[ ]= 1 2 3 is the com-

mand input vector (the user-defined levels of the three cli-mate variables). The variables z ii t, , , ,=1 2 3 areintegral-of-error states, introduced to ensure type 1 servo-mechanism performance (unity gain to the command inputsand steady-state decoupling: i.e., in the steady state, thecontrolled variables all reach their demand levels withoutany effect on the steady-state levels of the other output vari-ables).

PIP control system designs based on this NMSS model (2)provide the basis for any SVF design procedures, such as op-timal linear-quadratic (LQ), pole assignment, or risk-sensi-tive/robust design. In the simplest LQ case, for example,quadratic cost function weighting matricesQ = diag[ ]111111111 100 5 25 and R = diag [ . . ]1 01 01produce the SVF control law below (equation u t at the bottomof the next page), where the off-diagonals of the last 3 × 3block in the matrix have been constrained to zero consis-tent with the steady-state decoupling requirement. Notethat the pair [ , ]F G is stabilizable by the nonminimal statefeedback, and, in addition, we can always find a matrix Esuch that the Cholesky decomposition EE QT = , ensuringthat the pair [ , ]F E is observable [22]. Consequently, thisPIP-controlled system will be stable and achieve the desiredLQ optimality.

The results obtained when this controller is applied tothe model are shown by the fine lines in Fig. 2. The re-sponses show some coupling between the controlled vari-ables, with a 10% change in humidity resulting in atransient error of 1°C in the internal air temperature. Note



that the relative values of the cost function weighting ma-trix elements associated with the three integral-of-errorstates have been adjusted here to ensure that the rise

times of each channel have the required control character-istics, while minimizing harsh actuator movements thatcould cause wear.

The controller designed in the above manner was vali-dated in practical terms over a three-month implementationperiod during the 1993-1994 growing season with a tomatocrop in the greenhouse. Fig. 3 shows the control perfor-mance of each climate variable over the entire validation pe-riod. Each graph shows the percentage of the validationperiod that a control variable was inside a certain controllimit. For example, air temperature was less than 1°C awayfrom the set point for 98% of the validation period. This per-formance easily meets the requirements of the growers.

Finally, it should be noted that the results shown in Figs. 2and 3 were obtained with the simple, constant-gain,multivariable controller defined above, and some improve-ments in performance would be possible if more complex

control systems were used. For exam-ple, further decoupling of the tempera-ture and humidity control can beaccomplished by exploiting a specialform of multiobjective optimization de-veloped at Lancaster (e.g., [16]). Here,the Cholesky factors of the LQ costfunction weighting function matricesare optimized so that off-diagonalweightings are introduced into the costfunction to ensure satisfactory de-coupling (or other multiobjective re-quirements).

The responses obtained with suchan optimized PIP controller are shown

as the thick lines in Fig. 2; clearly, almost perfect decouplinghas been achieved. Other possible improvements include astability-guaranteed, neuro-fuzzy gain scheduling system


F =

−0 9056 0 0 0 0 0 0 0 0645 0 0 0 0

0 0 7906 0 0 0 0 0 0 6253 0 0 0 0

0 0 1

. .

. .

. . . .3537 0 0 0 4576 0 01171 581536 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0

− −

0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 9056 0 0 0 0 0 0 0 0645 0 1 0 0

0 0 7906 0 0 0 0 0 0 6253 0 0 1

− −−

. .

. . 0

0 0 13573 0 0 0 4576 0 01171 581536 0 0 1− −

. . . .

u t = −−22 8406 12938 0 0 0 0 0 0 6036 0 0001 5 6927 0 0

0 6954 0

. . . . .

. . . . .

. . .

9666 0 0 0 0 0 0 7149 0 0001 0 0 7991 0

0 0015 0 0021 0 0021

− −0 0 0 0055 0 0 0001 0 6996 0 0 0 0112− − − −

. . . .

x t

Temperature Humidity Carbon Dioxide

[ C]° [% RH] [p/m]

100 100 100

75 75 75

50 50 50

25 25 25

0 0 0

[%T

ime]

[%T

ime]

[%T

ime]

0 0.5 1 1.5 0 2 4 6 0 20 40 60

Figure 3. Long-term controller performance for the SRI Venlo greenhouse; presented aspercentage of total evaluation period (% time) against absolute control error in relation tothe set point (control limit).

u1 to y1

u1 to y2 u2 to y2

u2 to y11.5

1

0.5

0

0

−0.5

−1

−1.50 20 40

0 20 40

0

−0.05

−0.10 20 40

1.5

1

0.5

00 20 40

Figure 2. Closed-loop step responses for standard PIP-LQ (thincurves) and decoupling PIP-LQ (thick curves) controllers.


(e.g., [23]) or a fully adaptive PIP control system (e.g., [24]).The latter would be straightforward to implement, since therecursive estimation algorithms used in the modeling stageof the design can be easily implemented online in real time.However, the performance shown in Fig. 3 is perfectly ac-ceptable for most practical purposes, and such added com-plexity is not warranted in this case.

Modeling and Controlof Forced Ventilation SystemsThis example is related to the previous one and concerns theanalysis of data from planned experiments in a large instru-mented chamber in the Laboratory for Agricultural BuildingsResearch at the Katholieke Universiteit Leuven (for details,see [25]). This chamber, shown in Fig. 4, has a volume of 9 m3

and has been designed to represent a scale model of a live-stock building or greenhouse for use in experimental re-search on forced ventilation and heating in agriculturalbuildings. The two variable inputs, ventilation rate (120-300m3/h) and heating element input (0-400 W), determine thedominant airflow pattern within the chamber. An envelopechamber or “buffer zone” is constructed around the testroom to minimize disturbance of the airflow by heat conduc-tion from the laboratory. A series of aluminum conductorheat sinks and steam generation from a water reservoir pro-vide the internal heat and moisture production to simulateanimal occupants. To gain information about the distributionof mass and energy in a quantitative manner, 36 temperaturesensors and 24 humidity sensors are positioned in a three-di-mensional (3-D) array within the chamber.

The agricultural engineering literature reports a great dealof research on the modeling of ventilated air spaces, much ofit based on deterministic simulation models. More recentdata-based modeling has been stimulated by the control vol-ume (CV) concept of Barber and Ogilvie [26]. For example,Berckmans et al. [25] use least-squares methods to estimatethe parameters in a simple conceptual CV model of an imper-fectly mixed 3-D airspace from variations in ventilation rateand heat supply, whereas Daskalov [27] develops a model formeasured temperature and humidity variations in a naturallyventilated pig building using discrete-time TF models. To ob-tain the DBM model, which can be considered a natural ex-tension of these earlier models, the continuous-time SRIVestimation algorithm is used first to identify the linear TF (or-dinary differential equation) model between the measuredtemperatures at the inlet, Ti, and the outlet, T. In the case ofthe response to a step increase of the ventilation rate from 80to 300 m3/h, the estimated model for the change in tempera-ture ∆T from the initial steady levels takes the form

∆ ∆T ts

s sT ti t( )

. .. .

( )= ++ +

+1959 0 0892962 01112

ξ(3)

where s is the differential operator and ξ t is the residualnoise. Fig. 5 compares the output of this model with the mea-

sured change in the outlet temperature, and the associatedresidual ξ t is plotted in the lower graph. This unexplainedresidual has a zero mean value and low variance (0.021 com-pared with a measured variance of 2.619 for the output ∆Tseries; RT

2 0 992= . ).The TF model can be decomposed into a parallel or feed-

back connection of first-order processes, but the latter hasmore physical significance, as required in DBM modeling.This becomes clear if we invoke the classical theory of heattransfer and formulate the differential equations of dynamicheat transfer, bearing in mind the need to arrive at a sec-ond-order, lumped, differential equation with a TF similar to(3). Such analysis [28] yields the following equations for themain chamber and the buffer zone:

Main Chamber:d T

dtT K T Ti

( )∆ ∆ ∆ ∆= + −β α1 1 1buff

Buffer Zone:d T

dtK T K T T

( )( )

∆ ∆ ∆ ∆buffbuff buff= − + −2 3 .

In these equations,

β 11

= Vvol

; αγ1

1

1 1

1 1 1

= +

V kvol

sfvol cp

; Kk

11 1

1 1 1

= sfvol cpγ

;

Kk

22 2

2 1 1

= sfvol cpγ

and

Kk

31 1

2 1 1

= sfvol cpγ

where T is the temperature (°C) measured at the outletport but assumed to be representative of the temperaturein the active mixing volume (AMV) [10];Ti is the temperature


Figure 4. Large instrumented chamber in the laboratory atKatholieke Universiteit Leuven.


(°C) measured at the chamber inlet;V is the ventilation rate(m s3 1− ); γ1 is the air density of the incoming air and the air inthe main chamber (kgm 3− );cp1 is the specific heat capacity ofthe incoming air (Jkg C- -1 1° ) and the air in the main cham-ber; k1 is the thermal conduction coefficient through the in-ner chamber walls (Wm C2 1− −° ); sf1 is the surface area of theAMV for imperfect mixing (m 2); Tbuff is the temperature inthe outer buffer zone (°C); vol2 is the volume of the AMV as-sociated with the buffer zone (m 3); k2 is the thermal con-duction coefficient through the buffer zone (outerchamber) wall (Wm C2 1− −° ); andsf2 is the surface area of thebuffer zone AMV for imperfect mixing (m 2).

The differential equations above may be combined andexpressed in TF form to yield the following second-order TFmodel for the complete chamber-buffer zone system:

∆ ∆Tb s b

s a s aTi= +

+ +0 1

21 2 (4)

where

b0 1= β ; b K K1 1 2 3= +β ( ); ( )a K K1 2 3 1= + + α ;a K K K K2 2 3 1 1 3= + −( ) α .

Clearly, the TF model (4) has exactly the same structuralform as the estimated TF model (3), so in DBM modelingterms, it can be considered as one particular physical inter-pretation of the data-based model that has scientific credi-bility. The first-order TF numerator (B jj , ,= 0 1) anddenominator (A jj , ,= 0 1) parameters in the feedback decom-position can now be calculated as follows:

B0 1= β ; AV k

0 11

1 1

1 1 1

= = +

α

γvolsf

vol cp;

BK K k

V11 3

1

12

12

2 12

12

= =β γ

sfvol cp

;

A K Kk k

1 2 32 2

2 1 1

1 1

2 1 1

= + = +( )sf

vol cpsf

vol cpγ γ.

These two first-order TF’s have very different timeconstants: Tc1, associated with the A0 parameter, re-flects the fast response within the main chamber:

TA

Vk

c10

1

1 1

1 1

1 1

1= =

+

vol

sfcpγ

and Tc2, associated with the A1 parameter and themuch slower temperature effect due to feedbackconduction from the buffer zone, i.e., T Ac2 11= / .

The above analysis is useful because it showshow the inductive approach of DBM modeling objec-tively defines the lowest order structure (dynamicorder and feedback nature) of the model prior to de-fining a mechanistic interpretation compatible with

this structure. Note also that the conventionally definedheat transfer parameters used in the above mechanistic for-mulation of the equations are not identifiable from theabove relationships. In contrast, the parameters of themodel (4) have direct physical significance, and they can beidentified fully from the measurements obtained in thechamber experiments. Consequently, this model is not onlyuseful in subsequent control system design terms, but italso provides an alternative, mechanistic characterizationof the heat transfer dynamics that is useful in its own right.

The above model is currently being used in the design of amultivariable PIP controller for a chamber similar to that de-scribed in the previous section. However, other associatedresearch has been concerned with the modeling and controlof the fans used in the forced ventilation systems. Fig. 6, forexample, is a diagram of the fan test installation at Leuven.The model for this system is identified by the SRIV algorithmas a first-order discrete time TF, and this model has been usedas the basis for PIP and GPC system design. This study con-firms that the model-based control algorithms offer betterperformance than the conventional PID controller in terms ofboth the regulation of ventilation rate and the reduction ofenergy consumption. In particular, for a 450-mm axial fan, thenormalized mean square errors are as follows: PIP = 1, GPC =1.64, PID = 5.59. These results were obtained for an airflowrate of 3000 m3/h with realistic wind disturbances, but, in or-der to evaluate their practical robustness, all three control-lers were optimized for an airflow rate set point in theneighborhood of 1500 m3/h, without wind.

Environmental ForecastingPrevious publications from Lancaster illustrate how themodeling and forecasting procedures outlined above havebeen applied to environmental and other time series (e.g.,[13] and the references therein). Here, we will consider theresults obtained in the case of two practically important en-


2

0

−2

−4

−6

−8

−1

0

0

0

100

100

200

200

300

300

400

400

500

500

600

600

700

700

800

800

Tem

pera

ture

[C

]°

Outlet Temperature Response and Residuals

Time [Tens of Seconds]

Figure 5. Comparison of the model estimated temperature and themeasured temperature at the chamber outlet.


vironmental examples: first, the famousseries of monthly atmospheric CO2 mea-surements at Mauna Loa in Hawaii, asshown in Fig. 7, which did much to pro-mote the current debate on global warm-ing; and second, the hourly rainfall andriver flow series from the River RibbleCatchment in northwest England, asshown in Fig. 8.

Adaptive Forecasting,Backcasting,and Interpolation of theMauna Loa CO2 SeriesIt is clear from Fig. 7 that the variations inatmospheric CO2 at Mauna Loa are domi-nated by a long-term, upward trend and an-nual periodicity. As a result, the identifiedUC model takes the form

y T C et t t t= + + (5)

where

T a a t a t d C a t bt t t j tj

j j t= + + + = +=∑0 1 2

2

1

2

and , ,cos( ) (sinω ω j t )

(6)

and yt denotes theCO2. As shown, the trendTt is identified asa polynomial in time t, with unknown but constant parame-ters, combined with a residual signal dt that models the me-dium-term, stochastic deviations about this polynomialtrend. The annual seasonality is modeled by the periodiccomponent Ct : this is of a standard trigonometric (Fourier)form but with stochastically time-varying parameters (TVPs)a b jj t j t, ,, , ,=1 2. This type of relationship will model any peri-odic behavior defined by the frequencies ω j j, ,=1 2, and theTVPs allow for the estimation of changes in the amplitudeand phase of these seasonal variations. In the present CO2

example, the frequency ω π1 2 12= / is the fundamental fre-quency associated with the 12-month seasonal cycle, andω π2 2 6= / is its first harmonic. The spectrum of the data sug-gests that the other harmonics are insignificant and can beomitted. The TVPs dt , a b jj t j t, ,, , ,=1 2 in this UC model areidentified and modeled as simple random walk processes inan associated set of stochastic state equations (see [13]).The hyper-parameters (e.g., noise variance ratios) in thisstate-space model are estimated by a special form of optimi-zation in the frequency domain (see [12] and [13]) and therecursive KF and FIS algorithms are then used for forecast-ing, backcasting, and interpolation of the series.

Typical results are shown in Figs. 9 and 10. For this analy-sis, the estimation data set consisted of the first 313 samples(1958(1)-1985(5)) of the 504 data set, but with two years ofthese data between 1971(8) and 1973(8) omitted to showhow the algorithms interpolate over such a gap. For valida-


ControlFan

AirOut

6

12

13

10

11

8

914

7

5

431

2

VoltageController

DifferentialPressureSensor

ReferencePressureSensor

Pump

Data Acquisition

Measurementand Control Unit

ValvePosition

PressureChamber

Ventilation RateSensor

DisturbanceFan

VoltageController

Computer Control

PowerUnit

AirflowRate

Figure 6. Schematic layout of the test chamber and associated control equipment.

2.5

2

1.5

1

0.5

00

0

50

50

100

100

150

150

200

200

250

250

300

300

350

350

400

400

450

450

Flo

w

Rainfall-Flow Data for River Hodder at Hodder Place

6

4

2

0

Rai

nfal

l

Time [h]

Figure 8. Hourly rainfall (lower panel) and flow (upper panel)series for the River Hodder at Hodder Place in the River RibbleCatchment of northwest England.

370

360

350

340

330

320

3101960 1965 1970 1975 1980 1985 1990 1995 2000

Car

bon

Dio

xide

[ppm

]

Date

Atmospheric Carbon Dioxide at Mauna Loa, Hawaii, 1958-2000

Figure 7. Atmospheric CO2 measured at Mauna Loa, Hawaii,1958-2000.


tion purposes, true multistep-ahead forecasting is carriedout in the short term (three years ahead) and long term(14.4 years ahead to 1999(12)), respectively, based only onthe estimation data set, with the model optimized on the ba-sis of these same limited data (306 samples). The top panelin Fig. 9 shows the two-year interpolation results, whereasthe lower panel shows the three-year-ahead forecasts. Inboth cases, the results are excellent, with very small errorsbetween the interpolates/forecasts (full lines) and the data(dashed lines, not used in the analysis). The estimated stan-dard error bounds (95% confidence intervals) are shown asdotted lines. Fig. 10 shows the last 5.5 years of the14.4-year-ahead forecast. Here, the data (again not used inthe estimation or forecasting) are plotted as circular pointsand the forecast as a solid line, with the standard error

bounds shown dotted. Given the very longforecasting interval, these results are quiteremarkable and show how predictable thisimportant series can be if sufficiently pow-erful forecasting algorithms are used. As faras the authors are aware, these are the bestforecasting results produced so far for thisseries.

Adaptive Flow Forecastingin the Ribble CatchmentOnce again, the scientific literatureabounds with different models of rain-fall-flow processes in river catchments.These vary considerably in complexityfrom those based on continuum mechanics(solved approximately via finite differenceor finite element spatiotemporal discre-tization methods) [29] through conceptualmodels such as TOPMODEL [30] to hy-brid-metric-conceptual (HMC) grey-boxmodels such as IHACRES (e.g., [31]), whichare similar in structure and complexity tothe DBM model equivalent [32]. The mainaim of the River Ribble study [33] wasthreefold: first, to obtain DBM models relat-ing rainfall to the river flows measured inthe Ribble catchment; second, to evaluatethe performance of a state-adaptive,KF-based approach to forecasting, withthe state dynamics defined by these DBMmodels; and finally, to compare thisstate-adaptive approach with its predeces-sor at Lancaster, the parameter-adaptiveforecasting system developed for the Sol-way River Purification Board as a floodwarning system for the town of Dumfries inScotland (e.g., [34]). Another, secondaryobjective following from the development

of the Lancaster adaptive radar calibration (ARC) systemfor the National Rivers Authority/Environment Agency[35] was to consider how well forecasting performanceusing weather radar measured rainfall compared with theperformance using more conventional ground-basedmethods.

In the case of a uniform sampling interval of one hour,SRIV identification and estimation yields the following dis-crete-time TF model between the gauged rainfall rt and themeasured flow yt for the River Hodder at Hodder Place in theRibble catchment during December 1993:

yb b za z a z

g rt t t= ++ +

+−

− − −0 1

1

11

22 41

{ } ξ .(7)


335

330

325

1971 1971.5 1972 1972.5 1973 1973.5 1974 1974.5

Car

bon

Dio

xide

[ppm

]C

arbo

n D

ioxi

de [p

pm]

Interpolation and Forecasting Results

355

350

345

340

1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5Date

Figure 9. Two-year interpolation (upper panel) and three-year-ahead forecasting(lower panel) results for the Mauna Loa CO2 series.

380

375

370

365

360

355

3501995 1995.5 1996 1996.5 1997 1997.5 1998 1998.5 1999 1999.5 2000

Date

Car

bon

Dio

xide

[ppm

]

Long-Term 14 Year-Ahead Forecast, 1986-2000 (Last Five Years Shown)

Figure 10. Long-term, 14.4-year-ahead forecast of the Mauna Loa CO2 series,showing only the last five years for clarity.


Here, g rt{ }− 4 is a nonlinear function of the rainfall delayed byfour sampling intervals to introduce the pure (advective)time delay between the occurrence of the rainfall and itsfirst measured effect on the river flow. The noise level onflow data is normally quite large, and the noise term ξ t is in-troduced here to reflect the combined effects of measure-ment noise, unmeasured disturbances, and imperfectionsin the model.

In this case, SDP estimation shows that a rainfallnonlinearity is present and takes the form

u g r c f y rt t t t− − −= = ⋅ ⋅4 4 4{ } { } (8)

where ut is termed the “effective rainfall” (or “rainfall ex-cess”) and c is a scale factor chosen conventionally so thatthe volume of the effective rainfall is equal to the totalstream flow volume over the estimation period. Equation(8) shows that the rainfall affects the flow via a multiplica-tive nonlinearity between rt − 4 and a nonlinear function offlow f yt{ }, where yt is acting as a surrogate for the soil mois-ture, which is difficult to measure [9]. The estimated nonlin-ear function shows that the SDP is small for low flows (lowsoil moisture) and increases, but with diminishing slope, forhigh flows. Consequently, the effective rainfall is lower thanthe actual rainfall for low soil moisture, since it tends to beabsorbed by the dry soil under these conditions. At highflow (high soil moisture), however, the rainfall is much moreeffective in inducing flow variations.

Based on this SDP identification analysis and subsequentparametric estimation, the rainfall-flow nonlinearity isparametrized using a power law f y yt t{ } = θ to approximatef yt{ }. The SRIV estimates of the associated parameters, aswell as the TF parameters, are shown in Table 1. This also in-cludes some parameters that are derived from decomposi-tion of the TF model into the following parallel connection oftwo first-order processes (e.g., [9]):

b b za z a z z z

0 11

11

22

1

11

2

211 1 1

++ +

=+

++

−

− − − −

βα

βα (9)

where α βi i iand , ,=1 2 are, respectively, the eigenvalues ofthe TF denominator and the associated residues in the par-tial fraction expansion of the TF. The latter define theweighting or “partitioning” associated with this decomposi-tion. It is clear from (8) and (9) that the effective rainfallut − 4

can be partitioned into two parallel pathways resulting intwo component flows that, when added together, yield thetotal measured river flow. In effect, therefore, the decompo-sition provides estimates of two “unobserved” (“latent” or“hidden”) flow states and their associated rainfall-flow dy-namics.

The estimates in Table 1(b) reveal that the two pathwayshave very different dynamics. The “quick-flow” pathway hasa total travel time (Ti + δ) of 7.5 hr, whereas the “slow-flow”pathway has a total travel time of 77 hr. The associated par-

tition percentages of 60% and 40%, respectively, suggestthat more of the effective rainfall affects the quick-flow path-way than the slow-flow pathway. Note, however, that if theuncertainty in the parameter estimates is taken into ac-count using MCS analysis (see previous discussion and [8]),then some of the derived parameters in Table 1 have wideconfidence intervals and non-Gaussian distributions. In par-ticular, and not surprisingly, the slow-flow dynamics aremuch more poorly defined than the quick-flow dynamics.

As we have stressed, an important aspect of DBM model-ing is the physical interpretation of the model. Given the de-rived parameters in Table 1, the most obvious physicalinterpretation of the model (9) is that the effective rainfall af-fects the river flow nonlinearly via two main pathways. First,the initial rapid rise in the hydrograph, following an instanta-neous rise in effective rainfall, derives from the quick-flowpathway, probably as the aggregate result of the many sur-face processes active in the catchment. The subsequent longtail in the recession of the hydrograph is associated mainlywith the slow-flow component, probably as the result of wa-ter displacement within the groundwater system.

For flow forecasting purposes, the complete catchmentmodel for the River Ribble is formulated in terms of rain-fall-flow models such as (7), together with flow-flow TF mod-els that route the flow down the river system. All of these TFmodels are then combined into a set of stochastic stateequations with a state vector x t . In the case of the rain-fall-flow models, such as (7), the associated state variablesare defined as the latent states defined by decompositionssuch as (9), and the rainfall inputs are the effective rainfallseries, such asut . The estimate $x t of the state vector x t and,from this, the recursive estimate $yt of the river flow yt can


Table 1. (a) Model parameter estimates(to three decimal places).

Parameter Estimates

$ . ( . ); $ . ( . )a a1 21 739 0 015 0 743 0 014= − =$ . ( . ); $ . ( . )b b0 10153 0 006 0149 0 005= = −

$ . ; $ . ; $ . ; $ .α α β β1 2 1 20 986 0 753 0 006 0147= = = =

$ . ; $ . ( . )c = =0 847 0160 0 0001ϑ

(b) Derived physically meaningful parameters for (9).

PhysicallyMeaningful Derived

Parameters

Slow Flow Quick Flow

SSG, Gi 0.40 0.60

Residence time, Ti 73 hr 3.5 hr

Advective delay, δ 4 hr 4 hr

Partition %, pi 40% 60%


then be obtained from the associated KF algorithm (see[11], [33], and [36]), whose stochastic hyper-parameters(variance ratios) are estimated by maximum likelihood opti-mization based on prediction error decomposition [37]. Theresulting four-step-ahead forecasting results are obtainedfrom the prediction step in the KF update equations and areshown graphically in Fig. 11.

The forecasting performance shown in Fig. 11 can be im-proved a little if the noise ξ t in (7) is modeled as an AR pro-cess and introduced into the state-space description (thusincreasing its dimension to include the additional stochas-tic states). Even without this improvement, however, theperformance is good, and this is reflected in the statisticalproperties of the forecast errors, which have a mean valuenear zero (0.0032 mm⋅hr-1) and a standard deviationσe 4 0132= . mm⋅hr-1. The coefficient of determination based

on the variance of the four-step-ahead prediction errors isR4

2 0 871= . , compared with R42 0 484= . for the naïve (persis-

tence) forecast, where the predicted four-hour-ahead flow isequal to current flow.

ConclusionsThis article has briefly reviewed the main aspects of the ge-neric DBM approach to modeling stochastic dynamic sys-tems and shown how it is being applied to the analysis,forecasting, and control of environmental and agriculturalsystems. The advantages of this inductive approach to mod-eling lie in its wide range of applicability. It can be used tomodel linear, nonstationary, and nonlinear stochastic sys-tems, and its exploitation of recursive estimation meansthat the modeling results are useful for both online andoffline applications. To demonstrate the practical utility ofthe various methodological tools that underpin the DBM ap-

proach, the article also outlined several typical,practical examples in the area of environmental andagricultural systems analysis, where DBM modelshave formed the basis for simulation model reduc-tion, control system design, and forecasting. How-ever, the same methods have been used in manyother applications in diverse areas of science andsocial science. Recent applications include:

• Engineering: the delta operator modeling andautostabilization of the Harrier VSTOL aircraftin its most difficult transitional mode betweenhovering and normal flight [7], [16], the model-ing and control of interurban traffic systems[38], and both model reduction and control ofan industrial gasifier system [39].

• Ecology: the modeling of limit cycling behav-ior in blowfly population dynamics [7], [14].

• Biology: DBM modeling has revealed new as-pects of the dynamics associated withstomatal behavior in plants [40]; the chaoticelectrical activity in the axon of a squid hasalso been modeled [41], [42].

• Business Data Analysis and Forecasting: UC modelingand forecasting of microeconomic and business datafor the credit card company Barclaycard, U.K. [43]

• Macroeconomics: where DBM modeling yields somethought-provoking, and rather topical, insights intothe relationship between government spending, pri-vate capital expenditure, and unemployment in theUnited States between 1948 and 1998 [44].

AcknowledgmentsThe authors are grateful to the United Kingdom Biotechnol-ogy and Biological Sciences Research Council for its sup-port, through Research Grants 89/E06813 and 89/MMI09731,together with the Ph.D. studentship 98/B1/E/04226. The au-thors would also like to thank all the past and present mem-bers of the Environmental Systems and Control Group inCRES at Lancaster University who have contributed to theresearch described in this article, as well as Prof. DanielBerckmans and his staff in the Laboratory for AgriculturalBuildings Research at the Katholieke Universiteit Leuven,Belgium.

References[1] K.J. Beven, “Phrophecy, reality and uncertainty in distributed hydrologi-cal modelling,” Adv. Water Resources, vol. 16, pp. 41-51, 1993.[2] K. Keesman and G. Van Straten, “Set membership approach to identifica-tion and prediction of lake eutrophication,” Water Resources Research, vol. 26,pp. 2643-2652, 1990.[3] S. Parkinson and P.C. Young, “Uncertainty and sensitivity in global carboncycle modelling,” Climate Res., vol. 9, pp. 157-174, 1998.[4] K. Hasselmann, “Climate-change research after Kyoto,” Nature, vol. 390,pp. 225-226, 1997.[5] I. Seginer, “Some artificial neural network applications to greenhouse en-vironmental control,” Comput. Electron. Agriculture, vol. 18, pp. 167-186, 1997.


2.5

2

1.5

1

0.5

0

State-Adaptive Four-Step Ahead Forecasting: River Hodder at Hodder Place

0 50 100 150 200 250 300 350 400 450Time [h]

Flo

w [m

m/h

]

Figure 11. Four-step-ahead forecasting results for the River Hodder atHodder Place in December 1993: forecasts (solid line) compared withmeasured flow series (dashed).


[6] P.C. Young, S.D. Parkinson, and M. Lees, “Simplicity out of complexity inenvironmental systems: Occam’s Razor revisited,” J. Appl. Stat., vol. 23, no.2&3, pp. 165-210, 1996.[7] P.C. Young, “Data-based mechanistic modelling of environmental, ecologi-cal, economic and engineering systems,” Environ. Modeling Software, vol. 13,pp. 105-122, 1998.[8] P.C. Young, “Data-based mechanistic modelling, generalized sensitivity anddominant mode analysis,” Comput. Phys. Commun., vol. 115, pp. 1-17, 1999.[9] P.C. Young, “Time variable and state dependent parameter modelling ofnonstationary and nonlinear time series,” in Developments in Time Series, T.S.Rao, Ed. London: Chapman and Hall, 1993, pp. 374-413.[10] P.C. Young and M.J. Lees, “The active mixing volume (AMV): A new con-cept in modelling environmental systems,” in Statistics for the Environment, V.Barnett and K.F. Turkman, Eds. Chichester: Wiley, 1993, pp. 3-44.[11] P.C. Young, Recursive Estimation and Time-Series Analysis. Berlin:Springer-Verlag, 1984.[12] P.C. Young, “Nonstationary time series analysis and forecasting,” Prog-ress Environ. Sci., vol. 1, pp. 3-48, 1999.[13] P.C. Young, D. Pedregal, and W. Tych, “Dynamic harmonic regression,” J.Forecasting, vol. 18, pp. 369-394, 1999.[14] P.C. Young, “Stochastic, dynamic modelling and signal processing: Timevariable and state dependent parameter estimation,” in Nonstationary andNonlinear Signal Processing, W.J. Fitzgerald, A. Walden, R. Smith, and P.C.Young, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2000, pp. 74-114.[15] J.-S.R Jang, C.-T. Sun, and E. Mizutani, Neuro-Fuzzy and Soft Computing,Upper Saddle River, NJ: Prentice-Hall, 1997.[16] A. Chotai, P.C. Young, P. McKenna, and W. Tych, “Propor-tional-integral-plus design for delta operator systems: Part 2, MIMO sys-tems,” Int. J. Contr., vol. 70, no. 1, pp. 149-168, 1998.[17] C.J. Taylor, A. Chotai, and P.C. Young, “State space control system designbased on nonminimal state-variable feedback: Further generalisation andunification results,” Int. J. Contr., vol. 73, no. 14, pp. 1329-1345, 2000.[18] Z.S. Chalabi and B.J. Bailey, “Simulation of the energy balance in a green-house,” Div. Note, DN1516. AFRC Silsoe Research Institute, Bedford, 1989.[19] A.J. Udink ten Cate, “Modelling and simulation in greenhouse climatecontrol,” Acta Horticulturae, vol. 174, pp. 461-467, 1985.[20] G. Van Straten, H. Challa, and F. Buwalda, “Towards user accepted opti-mal control of greenhouse climate,” Comput. Electron. Agriculture, vol. 26, pp.221-238, 2000.[21] M. Lees, P.C. Young, A. Chotai, and W. Tych, “A nonminimal state variablefeedback approach to multivariable control of glasshouse climate,” Trans.Inst. Measure. Control, vol. 17, pp. 200-211, 1995.[22] B.C. Kuo, Digital Control Systems. New York: Holt, Rinehart and Winston,1980.[23] S.G. Cao, N.W. Rees, and G. Feng, “Stability analysis and design for a classof continuous-time fuzzy control systems,” Int. J. Contr., vol. 64, pp. 1069-1087,1996.[24] A. Chotai, P.C. Young, and M.A. Behzadi, “Self-adaptive design of a nonlin-ear temperature control system,” Proc. Inst. Elec. Eng., vol. 138, pt. D, pp.41-49, 1991.[25] D. Berckmans and M. De Moor, “Analysis of the control of livestock envi-ronment by mathematical identification on measured data,” in Proc. Int. Win-ter Meeting Ameri. Soc. Agric. Eng., Chicago, IL, 1993, paper no. 934574.[26] E.M. Barber and J.R. Ogilvie, “Incomplete mixing in ventilated spaces:Part 1. theoretical considerations,” Canadian Agric. Eng., vol. 24, pp. 25-29;Part 2. scale model study, vol. 26, pp. 189-196, 1984.[27] P.I. Daskalov, “Prediction of temperature and humidity in a naturally ven-tilated pig building,” J. Agric. Eng. Res., vol. 68, pp. 329-339, 1997.[28] L.E. Price, P.C. Young, D. Berckmans, K. Janssens, and C.J. Taylor,“Data-based mechanistic modelling and control of mass and energy transferin agricultural buildings,” Annu. Rev. Contr., vol. 23, pp. 71-82, 1999.

[29] M.B. Abbott, J.C. Bathurst, J.A. Cunge, P.E. O’Connell, and J.L. Rasmus-sen, “An introduction to the European Hydrology System (SHE). 2: Structureof a physically based distributed modelling system,” J. Hydrology, vol. 87, pp.61-77, 1986.[30] K.J. Beven and M.J. Kirkby “A physically-based variable contributingarea model of basin hydrology,” Hydrological Sci. J., vol. 24, pp. 43-69, 1979.[31] A.J. Jakeman and G.M. Hornberger, “How much complexity is warrantedin a rainfall-runoff model?” Water Resources Res., vol. 29, pp. 2637-2649, 1993.

[32] P.C. Young, “Data-based mechanistic modelling and validation of rain-fall-flow processes,” in Model Validation: Perspectives in Hydrological Science,M.G. Anderson and P.D. Bates, Eds. Chichester: Wiley, 2001, pp. 117-161.[33] P.C. Young and C. Tomlin, “Data-based mechanistic modelling and adap-tive flow forecasting,” in Flood Forecasting: What Does Current Research Offerthe Practitioner? M.J. Lees and P. Walsh, Eds. British Hydrological Society(BHS) Occasional Paper No. 12, Wallingford, U.K., 2000, pp. 26-40.[34] M.J. Lees, P.C. Young, K.J. Beven, S. Ferguson, and J. Burns, “An adaptiveflood warning system for the River Nith at Dumfries,” in River Flood Hydrau-lics, W.R. White and J. Watts, Eds. Wallingford, U.K.: Institute of Hydrology,1994, pp. 65-75.[35] M.E. Lord, P.C. Young, and R.C. Goodhew, “Adaptive radar calibration us-ing rain gauge data,” in Hydrological Uses of Weather Radar, British Hydrologi-cal Society Occasional Paper No.5. London: British Hydrological Society,1995.[36] R.E. Kalman, “A new approach to linear filtering and prediction prob-lems,” Trans. ASME, Journal Basic Eng., vol. 80, pt. D, pp. 468-478, 1960.[37] F. Schweppe, “Evaluation of likelihood functions for Gaussian signals,”IEEE Trans. Inform. Theory, vol. IT-11, no. 1, pp. 61-70, 1965.[38] C.J. Taylor, P.C. Young, A. Chotai, and J. Whittaker, “Nonminimal statespace approach to multivariable ramp metering control of motorway bottle-necks,” Proc. Inst. Elec. Eng., vol. 145, no. 6, pp. 568-574, 1998.[39] C.J. Taylor, A.P. McCabe, P.C. Young, and A. Chotai, “Propor-tional-integral-plus (PIP) control of the ALSTOM gasifier problem,” Proc. Inst.Mech. Eng., Journal of Systems and Control Engineering, vol. 214, pp. 469-480,2000.[40] A.J. Jarvis, P.C. Young, C.J. Taylor, and W.J. Davies, “An analysis of the dy-namic response of stomatal conductance to a reduction in humidity overleaves of Cedrella odorata,” Plant Cell and Environment, vol. 22, pp. 913-924,1999.[41] P.C. Young, “The identification and estimation of nonlinear stochasticsystems,” in Nonlinear Dynamics and Statistics, A.I. Mees, Ed. Boston:Birkhauser, 2001, pp. 127-166.[42] P.C. Young, P. McKenna, and J. Bruun, “Identification of nonlinear sto-chastic systems by state dependent parameter estimation,” Int. J. Contr., spe-cial issue on Nonlinear Identification and Estimation, in press, 2001.[43] W. Tych, D.J. Pedregal, and P.C. Young, “An unobserved component modelfor multi-rate forecasting of telephone call demand,” Int. J. Forecasting, 2001.[44] P.C. Young and D. Pedregal, “Macro-economic relativity: Governmentspending, private investment and unemployment in the USA 1948-1998,”Structural Change and Economic Dynamics, vol. 10, pp. 359-380, 1999.

Peter Young is Professor of Environmental Systems and Di-rector of the Centre for Research on Environmental Systemsand Statistics at Lancaster University. He is also Adjoint Pro-fessor of Environmental Systems at the Australian NationalUniversity, Canberra. He obtained the B.Tech. and M.Sc. de-grees at Loughborough University and the Ph.D. degree atCambridge University, where he was a Fellow of Clare Hall.His most recent research has been concerned withdata-based modeling, forecasting, signal processing, andcontrol for nonstationary and nonlinear stochastic systems.

Arun Chotai is a Senior Lecturer in the Environmental Sci-ence Department at Lancaster University and joint Head ofthe Systems and Control Group in the Centre for Researchon Environmental Systems and Statistics. He holds a Ph.D.in systems and control and a B.Sc. in mathematics, bothfrom the University of Bath. His research interests are inmodeling and control design, including self-adaptive,self-tuning, and multivariable control of nonstationary andnonlinear systems.



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